The linear, rotating and odd-viscous shallow-water model describes a two-dimensional thin layer of fluid of height \(\eta \)  and velocity \((u,v)\) . Such variables are ruled by a system of linear partial differential equations that can be rewritten as a Schrödinger equation [1]}, [2]} :
\({\mathrm {i}}\partial _t \psi = \mathcal {H} \psi , \qquad \psi = \begin{pmatrix}\eta \\ u\\ v\end{pmatrix}, \qquad \mathcal {H} = \begin{pmatrix}0 & p_x & p_y \\ p_x & 0 & -{\mathrm {i}}(f-\nu p^2) \\ p_y & {\mathrm {i}}(f-\nu p^2) & 0\end{pmatrix},\) 
